# Sharp Spectral Thresholds for Logit Fixed Points

arXiv: https://arxiv.org/abs/2605.15651 · HTML: https://mlsanity.com/paper/2605.15651/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in RL theory and sample complexity, one of 15 directions in Deep reinforcement learning methods.
- Builds on 9 of its 48 references that are in the map.
- First posted on arXiv 15 May 2026.

## Placed in

- Deep reinforcement learning methods / [RL theory and sample complexity](https://mlsanity.com/deeprl/rl-theory-and-sample-complexity/)

## Builds on (most cited first, 9 of 9)

- [A Theory of Regularized Markov Decision Processes](https://mlsanity.com/paper/1901.11275/) (2019)
- [A unified view of entropy-regularized Markov decision processes](https://mlsanity.com/paper/1705.07798/) (2017)
- [Fast Global Convergence of Natural Policy Gradient Methods with Entropy Regularization](https://mlsanity.com/paper/2007.06558/) (2020)
- [On the Properties of the Softmax Function with Application in Game Theory and Reinforcement Learning](https://mlsanity.com/paper/1704.00805/) (2017)
- [Beyond Exact Gradients: Convergence of Stochastic Soft-Max Policy Gradient Methods with Entropy Regularization](https://mlsanity.com/paper/2110.10117/) (2021)
- [Softmax is 1/2-Lipschitz: A tight bound across all ℓp norms](https://mlsanity.com/paper/2510.23012/) (2025)
- [Positive feedback in coordination games: stochastic evolutionary dynamics and the logit choice rule](https://mlsanity.com/paper/1701.04870/) (2017)
- [On the stability of the logit dynamics in population games](https://mlsanity.com/paper/2307.03507/) (2023)
- [An Analysis of Logit Learning with the r-Lambert Function](https://mlsanity.com/paper/2409.05044/) (2024)

## Abstract (arXiv)

Softmax feedback systems are a common mathematical core of entropy-regularized reinforcement learning, logit game dynamics, population choice, and mean-field variational updates. Their central stability question is simple: when does a self-reinforcing softmax system produce a unique and globally predictable outcome? Classical theory gives a conservative answer. By treating softmax as a unit-scale response, it certifies stability only in a strongly randomized regime. We prove that the classical approach misses an entire stable regime and does not identify the point at which the qualitative change truly occurs. For finite-dimensional affine logit systems, the sharp dimension-free Euclidean threshold is β|Pi WPi|mathcal T→mathcal T<2, rather than the previously used condition, which certifies stability only while the softmax system remains safely over-regularized. Our theorem fills the previously missing pre-bifurcation regime, extending stability guarantees for affine softmax feedback systems to reward-responsive yet globally predictable systems. It enlarges the certified stability boundary for these systems and identifies where the model genuinely undergoes a phase transition.